1. Complexities for the Design of Self-Assembly Systems
May 3, 2006
Robert Schweller Electrical Engineering and Computer Science Department Northwestern University

2. Outline Background, Motivation Model Temperature Programming
Flexible Glue Self-Assembly
Shape Verification
Future Work

3. Molecular Building Blocks
T G C G
A C G C

4. Molecular Building Blocks
[John Reif’s Group,
Duke University]

5. DNA Scaffolding
[Sung Ha Park, Constantin Pistol, Sang Jung Ahn, John H. Reif, Alvin R. Lebeck, Chris Dwyer, and Thomas H. LaBean, 2006]

6. Simulation of Cellular Automata
Paul Rothemund, Nick Papadakis, Erik Winfree, PLoS Biology 2: e424 (2004)
340nm

7. Outline Background, Motivation Model Temperature Programming
Shape Verification
Flexible Glue Self-Assembly
Future Work

8. Tile Model of Self-Assembly
(Rothemund, Winfree STOC 2000)
Tile System:
t : temperature, positive integer
G: glue function
T: tileset
s: seed tile

9. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

10. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

11. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

12. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

13. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

14. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

15. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

16. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

17. How a tile system self assembles
G(y) = 2
G(g) = 2
G(r) = 2
G(b) = 2
G(p) = 1
G(w) = 1
t = 2
T =

18. Each Shape Requires a Distinct Tile Set

19. Programmable, General Purpose Tile Set?

20. Programmable, General Purpose Tile Set?
. . .

21. Outline Background, Motivation Model Temperature Programming
Flexible Glue Self-Assembly
Shape Verification
Future Work

22. Multiple Temperature Model
- temperature may go up and down

23. Multiple Temperature Model
- temperature may go up and down t
< t1 , t2 , ... , tr-1 , tr >

24. Multiple Temperature Model
- temperature may go up and down t
< t1 , t2 , ... , tr-1 , tr >
Tile Complexity: Number of Tiles
Temperature Complexity: Number of Temperatures

25. Building k x n Rectangles
k-digit, base n(1/k) counter: k n

26. Building k x n Rectangles
k-digit, base n(1/k) counter: k
If N is the kth power of some integer, then you choose a base that is big enough and then seed the counter to an appropriate value.
Note that for k<<N, N^1/k dominates. n
Tile Complexity:

27. two temperatures
t = 4 3 1 3 3 n

28. two temperatures
t = n

29. two temperatures
Tile Complexity:
t = n

30. two temperatures Tile Complexity: n Kolmogorov Complexity
(Rothemund,
Winfree STOC 2000)
Beats Standard Model

31. Programmable, General Purpose Tile Set?
. . .

32. High Level Approach
Given: n
log n

33. High Level Approach
Given: n
log n 1
temp

34. High Level Approach
Given: n
log n 1
temp 1

35. High Level Approach
Given: n
log n 1
temp 1

36. High Level Approach Given: n 1011001 log n 1 1 1 . . . . . . temp 1 11 1
. . .
. . .
temp 1 1 1 1

37. High Level Approach
. . . 1
. . .
temp 1 1 1 1

38. High Level Approach
. . . 1
. . .
temp 1 1 1 1

39. High Level Approach
. . . 1
. . .
temp 1 1 1 1

40. Assembly of n x n Squares
N - k k

41. Assembly of n x n Squares
n - k k

42. Assembly of n x n Squares
n - k k

43. Assembly of n x n Squares
n - k
Complexity: k

44. Assembly of n x n Squares
n – log n
Complexity:
log n

45. Assembly of n x n Squares
n – log n
Complexity:
seed row
log n

46. Encoding a Single Bit 1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g

47. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

48. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

49. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

50. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

51. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

52. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 1 z Z g g g g g g g g a g g a

53. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 1 z Z g g g g g g g g a g g a

54. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z z Z g g g g g g g g1 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

55. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z 0’ z z Z g g g g g g1 1 0’ 1’ z z 1 0’ z z Z g g g g g g g g a g g a

56. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z 0’ z Z g g Z g g g g1 1 0’ 1’ z z 1 0’ z Z g g Z g g g g g g a g g a

57. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z 0’ z Z g g Z g g g g1 1 0’ 1’ z z 1 0’ z Z g g Z g g g g g g a g g a

58. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z 0’ z Z g g Z g g g g1 1 0’ 1’ z z 1 0’ z Z g g Z g g g g g g a g g a

59. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z 0’ z Z g g Z g g g g1 1 0’ 1’ z z 1 0’ z Z g g Z g g g g g g a g g a

60. Encoding a Single Bit t = < 2 > 1 0’ 1’ z z 0’ z Z g g Z g g g g1 1 0’ 1’ z z 1 0’ z Z g g Z g g g g g g a g g a

61. Encoding a Single Bit t = < 2, 5 > 1 0’ 1’ z z 0’ z Z g g Z g g1 1 0’ 1’ z z 1 0’ z Z g g Z g g g g g g a g g a

62. Encoding a Single Bit t = < 2, 5 > x 1 0’ 1’ z z 0’ z Z g g Z g1 1 0’ 1’ z z 1 x 0’ z Z g g Z g g g g g g a g g a

63. Encoding a Single Bit t = < 2, 5 > 1 0’ 1’ 0’ z z z Z g g Z g g1 1 0’ 1’ 0’ z z 1 1 z Z g g Z g g g g g g a g g a

64. Encoding a Single Bit t = < 2, 5 > 1 0’ 1’ 1’ z z z z Z g g Z g1 1 0’ 1’ 1’ z z 1 1 z z Z g g Z g g g g g g a g g a

65. Encoding a Single Bit t = < 2, 5 > 1 0’ 1’ z z 1’ z Z g g Z g g1 1 0’ 1’ z z 1 1’ z Z g g Z g g g g g g a g g a

66. Encoding a Single Bit t = < 2, 5 > 1 0’ 1’ 1 z z 1’ z Z g g Z g1 1 0’ 1’ 1 z z 1 1’ z Z g g Z g g g g g g a g g a

67. Encoding a Single Bit t = < 2, 5 > t = < 2 > 1 0’ 1’ 1 z z1 1 0’ 1’ 1 z z 1 0’ 1’ z Z g g Z Z g g g g g g a g g a a

68. Goal:

69. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > a s b

70. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a s b

71. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a s X b

72. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a s Y b

73. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a a s Y b b

74. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a a s Y b b

75. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a a s Y X b b

76. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a a a s Y Y b b b

77. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 a a a s Y Y b b b

78. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a s Y Y b b b

79. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a s Y Y X b b b

80. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a a s Y Y Y b b b b

81. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a a s Y Y Y b b b b

82. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a a s Y Y Y X b b b b

83. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a a a s Y Y Y Y b b b b b

84. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a a a s Y Y Y Y b b b b b

85. Goal:
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 > 1 1 a a a a a s Y Y Y Y X b b b b b

88. Assembly of n x n Squares
n – log n
log n

89. Assembly of n x n Squares
O(log n)

90. Assembly of n x n Squares
O(log n)

91. Results O(1) n x n squares O(log n) O(1) tile complexity
temperature complexity
Temperature
Programming
O(1)
O(log n)
(Adleman, Cheng,
Goel, Huang
STOC 2001)
O(1)

92. Results O(1) n x n squares O(log n) O(1) ? < ? < log n
tile complexity
temperature complexity
Temperature
Programming
O(1)
O(log n)
(Adleman, Cheng,
Goel, Huang
STOC 2001)
O(1)
Smooth Trade off?
? <
? < log n

93. Results O(1) n x n squares O(log n) O(1) ? < ? < log n
tile complexity
temperature complexity
Temperature
Programming
O(1)
O(log n)
(Adleman, Cheng,
Goel, Huang
STOC 2001)
O(1)
Smooth Trade off?
? <
? < log n
For almost all n, no tileset can achieve both o(log n/ loglog n) tile complexity and o(log n) temperature complexity simultaneously

94. General Shapes? General Scaled Shapes [Soloveichik, Winfree 2004]
O(Ks*) Tile complexity, single temperature
Combined with Temperature Programming:
O(1) Tile Complexity
O(Ks) Temperature Complexity
General, Constant Scaled Shapes?
O(|S|) Temperature Complexity
Multiple temperature model
raising and lowering temperature multiple times
monotonically increasing temperatures
Time complexity versus tile complexity
multiple tile model and time complexity

95. Outline Background, Motivation Model and Basics
Temperature Programming
Flexible Glue Self-Assembly
Shape Verification
Future Work

96. Only identical glues attract
G(y,y) = 2
G(g,g) = 2
G(r,r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =

97. Only identical glues attract
G(y,y) = 2
G(g,g) = 2
G(r,r) = 2
G(b,b) = 2
G(p,p) = 1
G(w,w) = 1
t = 2
T =
We don’t have:
G(p,b) = 1

98. Flexible Glue Model Substantial Reduction in Tile Complexity
Remove the restriction that G(x, y) = 0 for x!=y
Substantial Reduction in Tile Complexity
Design of Flexible glues

99. n x n Squares --- Flexible Glue Model
Kolmogorov lower bounds:
Standard
(Rothemund, Winfree
STOC 2000)
Flexible
Standard Glue Function
Flexible Glue Function
a b c d e f a b c d e f
a b c d e f a b c d e f

100. Assembly of n x n Squares
n – log n
Complexity:
seed row
log n

101. n x n Square --- Flexible Glue Model
goal:
- seed binary counter to a given value - 1 1 1 1 1 1 1 1 1 1 1
All the complexity is coming from that damn seed row! 2
log n

102. n x n Square --- Flexible Glue Model5 3 3 3 4 4 4 4 4 4 5 5 5 5
. . . 3 4 5 1 2 3 4 5 1 2 3 4 5
All the complexity is coming from that damn seed row!

103. n x n Square --- Flexible Glue Model
key idea: 5
| | | | | | | | | | | | | 5 3 3 3 4 4 4 4 4 4 5 5 5 5
. . . 3 4 5 1 2 3 4 5 1 2 3 4 5
All the complexity is coming from that damn seed row!

104. n x n Square --- Flexible Glue Model
G(b4, p5) = 1
G(b4, w5) = 0 5 p5 5 5 5 5 w5 b4 1 2 3 4 5

105. n x n Square --- Flexible Glue Model5
given B = …
encode B into glue function p5 b4 4
p0 p1 p2 p3 p4 p5 b b b b b b
B = …

106. n x n Square --- Flexible Glue Model
build block
Complexity:

108. Assembly of n x n Squares
n – log n
2 x log n block
log n

109. Assembly of n x n Squares
Complexity:
O(log n)

110. Assembly of n x n Squares
Tile Complexity for
n x n squares
Flexible Glue:
(Adleman, Cheng, Goel, Huang
STOC 2001)
Standard:

111. Outline Background, Motivation Model and Basics
Temperature Programming
Flexible Glue Self-Assembly
Reduced Tile Complexity
Glue Design
Shape Verification
Future Work

112. Glue Design ACGGT TGCCA GGGAT CCCTA GTTGG CAACC CGTAC GCATG GACTC
CTGAG

113. Glue Design- Standard ACGGT TGCCA GGGAT CCCTA GTTGG CAACC CGTAC GCATG
GACTC
CTGAG

114. Glue Design- Standard ACGGT TGCCA GGGAT CCCTA GTTGG CAACC CGTAC GCATG
GACTC
CTGAG

115. Glue Design- Standard ACGGT
Design n strings such that each pair of strings has Hamming distance at least .
GGGAT
GTTGG
CGTAC
GACTC

116. Flexible Glue Design

117. Flexible Glue Design

118. Flexible Glue Design

119. Flexible Glue Design if (u,v)  G, HD((u), (v)) ≤ 
Input:
- Graph G(V,E)
- separation parameter 
Output:
A -labeling:
A labeling :V→{0,1}* such that for some , ,  -   ,
if (u,v)  G, HD((u), (v)) ≤ 
if (u,v)  G, HD((u), (v))  
Minimize label length

120. Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

121. Graph Decomposition Strategy: - Decompose graph - Label each Subgraph
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems
Strategy:
- Decompose graph
- Label each Subgraph
- Concatenate labels

122. Graph Decomposition Strategy: - Decompose graph - Label each Subgraph
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems
Strategy:
- Decompose graph
- Label each Subgraph
- Concatenate labels

123. Graph Decomposition Need Exact -labeling:
Each pair of non-adjacent nodes have
exactly the same Hamming distance.
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

124. Exact -labeling for Matchings
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

125. Exact -labeling for Matchings
Hadamard Code (n, )
Example: n=8, =4
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems
Hamming Distance:
Exactly 
Word Length: O(n + )

126. Exact -labeling for Matchings
Hadamard Code (n, )
Example: n=8, =4
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems
Hamming Distance:
Exactly 
Word Length: O(n + )

127. Matching Decomposition
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

128. Matching Decomposition
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

129. Matching Decomposition
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems
Length per Matching: O(n + )
Number of Matchings: O(D)
Label Length: O(Dn + D)

130. Tough Example Label Length: O(Dn + D) Length per Matching: O(n + )
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems
Length per Matching: O(n + )
Number of Matchings: O(D)
Label Length: O(Dn + D)

131. Star Graphs length O(n) exact -labeling
Star Graph: All edges are adjacent to the same vertex
- Non-trivial application of the Hadamard Code yields:
length O(n) exact -labeling
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

132. Star Destroyer 1. Decompose graph into “large” stars:
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

133. Star Destroyer 1. Decompose graph into “large” stars:
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

134. Star Destroyer 1. Decompose graph into “large” stars:
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

135. Star Destroyer 1. Decompose graph into “large” stars:
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
2. Decompose remaining graph into matchings:
- O(n+) length per matching
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

136. Star Destroyer Star Destoyer yields length:
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
2. Decompose remaining graph into matchings:
- O(n+) length per matching
Generalization over the standard coding theory problem where we want the hamming distance between all nodes to be large
Relationship with graph labeling problems

137. Flexible Word Design Results
Word Length
Matching Algorithm
General Graphs
Star Destoyer
Other Work
-Trees, Lines and Rings
-Distance Labeling
-Application to applying DNA computing to digital signal processing

138. Outline Background, Motivation Model and Basics
Temperature Programming
Flexible Glue Self-Assembly
Shape Verification
Future Work

139. Shape Verification Shape Verification Problem Input: T, a tile system
S, a shape
Question: Does T uniquely assemble S.
Standard: P (Adleman, Cheng, Goel, Huang, Kempe,
Flexible glue: P Espanes, Rothemund, STOC 2002)
Unique Shape: Co-NPC
Multiple Temperature: NP-hard, Co-NP-hard
Multiple Tile: NP-hard, Co-NP-hard

140. Shape Verification: Unique-Shape Model*

141. Shape Verification: Unique-Shape Model* x3 x2 x1 *

142. Shape Verification: Unique-Shape Model* x3 x2 x1 * * c1 c2 c3 *

143. * 1 x3 x2 x1 * * c1 c2 c3 * Shape Verification: Unique-Shape Model x xx x2 x x1 x * * c1 c2 c3 *

144. Shape Verification: Unique-Shape Model* x3 1 x2 1 x1 * * c1 c2 c3 *

145. Shape Verification: Unique-Shape Model* x3 1 x2 1 x1 c1 * * c1 c2 c3 *

146. Shape Verification: Unique-Shape Model* x3 1 x2 1 ok x1 c1 * * c1 c2 c3 *

147. Shape Verification: Unique-Shape Model* x3 1 ok x2 1 ok x1 c1 * * c1 c2 c3 *

148. Shape Verification: Unique-Shape Model* x3 1 ok x2 1 ok x1 c1 c2 * * c1 c2 c3 *

149. Shape Verification: Unique-Shape Model* x3 1 ok x2 1 ok c2 x1 c1 c2 * * c1 c2 c3 *

150. Shape Verification: Unique-Shape Model* x3 1 ok ok x2 1 ok c2 x1 c1 c2 * * c1 c2 c3 *

151. * x3 1 ok ok x2 1 ok c2 x1 c1 c2 ok * * c1 c2 c3 *
Shape Verification: Unique-Shape Model * x3 1 ok ok x2 1 ok c2 x1 c1 c2 ok * * c1 c2 c3 *

152. * x3 1 ok ok ok x2 1 ok c2 ok x1 c1 c2 ok * * c1 c2 c3 *
Shape Verification: Unique-Shape Model * x3 1 ok ok ok x2 1 ok c2 ok x1 c1 c2 ok * * c1 c2 c3 *

153. * * x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *
Shape Verification: Unique-Shape Model * * x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *

154. * * T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *
Shape Verification: Unique-Shape Model * * T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *

155. * * T T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *
Shape Verification: Unique-Shape Model * * T T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *

156. * * T T T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *
Shape Verification: Unique-Shape Model * * T T T x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *

157. Satisfied * * T T T SAT x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok *
Shape Verification: Unique-Shape Model * * T T T
SAT x3 1 ok ok ok * x2 1 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

158. Satisfied * * T T T SAT * * x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2
Shape Verification: Unique-Shape Model * * T T T
SAT * * x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

159. Satisfied * * T T T SAT * * T x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2
Shape Verification: Unique-Shape Model * * T T T
SAT * * T x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

160. Satisfied * * T T T SAT * * T F x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok
Shape Verification: Unique-Shape Model * * T T T
SAT * * T F x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
(LaBean and Lagoudakis, 1999)

161. Not Satisfied Satisfied * * T T T SAT * * T F F x3 1 ok ok ok * x3 ok
Shape Verification: Unique-Shape Model * * T T T
SAT * * T F F x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied
(LaBean and Lagoudakis, 1999)

162. * T T T T * T T F T x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1
Shape Verification: Multiple Temperature Model * T T T T
SAT * T T F T NO x3 1 ok ok ok * x3 ok c2 ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied

163. * T T T T * T T T T x3 1 ok ok ok * x3 1 ok ok ok * x2 1 ok c2 ok * x2
Shape Verification: Multiple Temperature Model * T T T T
SAT * T T T T NO x3 1 ok ok ok * x3 1 ok ok ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied

164. * T T T T * T T T T x3 1 ok ok ok * x3 1 ok ok ok * x2 1 ok c2 ok * x2
Shape Verification: Multiple Temperature Model * T T T T
SAT * T T T T NO x3 1 ok ok ok * x3 1 ok ok ok * x2 1 ok c2 ok * x2 1 ok c2 ok * x1 c1 c2 ok * x1 c1 c2 ok * * * c1 c2 c3 * * * c1 c2 c3 *
Satisfied
Not Satisfied

165. * * x3 x3 x2 x2 x1 x1 * * Satisfied Not Satisfied
Shape Verification: Multiple Temperature Model * * x3 x3 x2 x2 x1 x1 * *
Satisfied
Not Satisfied

166. * * T F F x3 ok c2 ok * x2 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *
Shape Verification: Multiple Temperature Model * * T F F NO x3 ok c2 ok * x2 ok c2 ok * x1 c1 c2 ok * * * c1 c2 c3 *

167. Shape Verification: Multiple Temperature Model* * T F F NO * x3 ok c2 ok * x2 ok c2 ok * x1 c1 c2 ok * 1 * * c1 c2 c3 * *

168. Shape Verification: Multiple Temperature Model* * T F F NO * F F F NO x3 ok c2 ok * c1 c2 ok * x2 ok c2 ok * c1 c2 ok * x1 c1 c2 ok * 1 c1 c2 ok * * * c1 c2 c3 * * c1 c2 c3 *

169. ... Shape Verification: Multiple Temperature Model * * T F F * F F F *NO * F F F NO * NO * T T F NO x3 ok c2 ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *

170. ... Shape Verification: Multiple Temperature Model * * T F F * F F F *NO * F F F NO * NO * T T F NO x3 ok c2 ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *

171. ... Shape Verification: Multiple Temperature Model * * T F F * F F F *NO * F F F NO * NO * T T F NO x3 ok c2 ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *

172. ... ... Shape Verification: Multiple Temperature Model * * T F F * F FNO * F F F NO * NO * T T F NO x3 ok c2 ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 * * * T T T
SAT * F F F NO * NO * T T F NO x3 ok ok ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *

173. Not Satisfiable Satisfiable
Shape Verification: Multiple Temperature Model * x3
Not Satisfiable x2 x1 * * x3
Satisfiable x2 x1 *

174. Shape Verification: Multiple Temperature Model
Input Shape:

175. Input Shape: Shape Verification: Multiple Temperature Model ... * * TNO * F F F NO * NO * T T F NO x3 ok c2 ok * c1 c2 ok * * 1 ok ok c3 * x2 ok c2 ok * c1 c2 ok * 1
... * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *

176. Input Shape: Co-NP-hard Equivalent to Co-SAT
Shape Verification: Multiple Temperature Model
Input Shape: * * T F F NO * F F F NO * NO * T T F NO x3 ok c2 ok * c1 c2 ok * * 1 ok ok c3 * x2 ok c2 ok * c1 c2 ok * 1
... * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *
Co-NP-hard
Equivalent to Co-SAT

177. Shape Verification: Multiple Temperature Model
Input Shape:

178. Input Shape: Shape Verification: Multiple Temperature Model ... * * T
SAT * F F F NO * NO * T T F NO x3 ok ok ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *

179. Input Shape: Equivalent to SAT NP-hard
Shape Verification: Multiple Temperature Model
Input Shape: * * T T T
SAT * F F F NO * NO * T T F NO x3 ok ok ok * c1 c2 ok * * 1 ok ok c3 *
... x2 ok c2 ok * c1 c2 ok * 1 * 1 ok c2 c3 * x1 c1 c2 ok * 1 c1 c2 ok * * 1 c1 c2 c3 * * * c1 c2 c3 * * c1 c2 c3 * * * * c1 c2 c3 *
Equivalent to SAT
NP-hard

180. Shape Verification Results
Standard P
Flexible Glue P
Multiple Temperature NP-hard
Co-NP-hard
Unique Shape Co-NPC
Multiple Tile NP-hard
(Adleman, Cheng,
Goel, Huang, Kempe,
Espanes, Rothemund,
STOC 2002)

181. Outline Background, Motivation Model and Basics
Temperature Programming
Flexible Glue Self-Assembly
Shape Verification
Future Work

182. Shape Replication

183. Shape Replication
drop temperature

184. Shape Replication
drop temperature

185. Shape Replication
drop temperature
raise temperature

186. Shape Replication
drop temperature
raise temperature
drop temperature

187. Shape Replication

188. Shape Replication

189. Shape Replication
raise temperature

190. Shape Replication
raise temperature
drop temperature

191. Assembly takes place in stages, each stage with a different tile set
Staged Assembly
Assembly takes place in stages, each stage with a different tile set
Large drops in Tile Complexity
Filter-Free Staged Assembly
Can only add tiles, never remove
Still, large drops in Complexity

192. Future Work Lab Work Experimental tests for Temperature Programming
Flexible Glue Design

193. Thanks for Listening Questions?
Robert Schweller Electrical Engineering and Computer Science Department Northwestern University